{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:33:38Z","timestamp":1787337218509,"version":"build-2736575974"},"reference-count":44,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>The application of variable-step time-integration techniques in a method of lines approach for the numerical solution of transient elastohydrodynamic lubrication calculations is described. Spatial discretization of the coupled integro-differential equations leads to a system of differential-algebraic equations in time. At each timestep a large system of nonlinear equations of 104--107 variables is solved by using multigrid methods and multilevel multi-integration. The application of temporal error control methods is considered with reference to both the differential-algebraic formulation of the system to be solved and the multigrid-based nonlinear equation solver to be used. The error control method used takes account of the magnitude of the spatial error present by using a local time error control based on the magnitude of this error. This approach is shown to be beneficial in reducing the computational work required. Experimental results on a variety of lubrication problems, including a challenging rough surface problem, are used to display the effectiveness of the methods.<\/jats:p>","DOI":"10.1137\/050622092","type":"journal-article","created":{"date-parts":[[2008,2,6]],"date-time":"2008-02-06T16:45:19Z","timestamp":1202316319000},"page":"626-650","source":"Crossref","is-referenced-by-count":10,"title":["Adaptive Timestepping for Elastohydrodynamic Lubrication Solvers"],"prefix":"10.1137","volume":"28","author":[{"given":"C. E.","family":"Goodyer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Berzins","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0916036"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(90)90171-V"},{"key":"R3","volume-title":"Numerical solution of initial\u2010value problems in differential\u2010algebraic equations","author":"Brenan K.","year":"1996"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719505"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827503420969"},{"key":"R6","unstructured":"D. Dowson and G. R. Higginson,\n                      Elasto\u2010hydrodynamic Lubrication, The Fundamentals of Roller and Gear Lubrication\n                      , Permagon Press, Oxford, Great Britain, 1966."},{"key":"R7","unstructured":"D. Dowson and C. M. Taylor,\n                      Cavitation in bearings\n                      , in Annual Review of Fluid Mechanics Vol. 11, Annual Reviews, Palo Alto, CA, 1979, pp. 35\u201366."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1243\/1350650011543574"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0732001"},{"key":"R10","unstructured":"R. Fairlie, C. E. Goodyer, M. Berzins, and L. E. Scales,\n                      Numerical modelling of thermal effects in elastohydrodynamic lubrication solvers\n                      , in Tribological Research and Design for Engineering Systems, Proceedings of the 29th Leeds\u2010Lyon Symposium on Tribology, D. Dowson et al., eds., Elsevier, Amsterdam, 2003, pp. 675\u2013683."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1002\/fld.632"},{"key":"R12","unstructured":"R. Gohar,\n                      Elastohydrodynamics\n                      , 2nd ed., Imperial College Press, London, 2001."},{"key":"R13","unstructured":"C. E. Goodyer,\n                      Adaptive Numerical Methods for Elastohydrodynamic Lubrication\n                      , Ph.D. thesis, University of Leeds, Leeds, UK, 2001."},{"key":"R14","unstructured":"C. E. Goodyer, M. Berzins, P. K. Jimack, and L. E. Scales,\n                      Grid\u2010based numerical optimisation in a problem solving environment\n                      , in Proceedings of the All Hands Meeting 2003, S. Cox, ed., EPSRC, Swindon, UK, 2003, pp. 854\u2013861."},{"key":"R15","unstructured":"C. E. Goodyer, R. Fairlie, M. Berzins, and L. E. Scales,\n                      Adaptive mesh methods for elastohydrodynamic lubrication\n                      , in ECCOMAS CFD 2001:"},{"key":"R15","unstructured":"Computational Fluid Dynamics Conference Proceedings, Institute of Mathematics and Its Applications, Minneapolis, MN, 2001."},{"key":"R16","unstructured":"C. E. Goodyer, R. Fairlie, M. Berzins, and L. E. Scales,\n                      Adaptive techniques for elastohydrodynamic lubrication solvers\n                      , in Tribology Research: From Model Experiment to Industrial Problem, Proceedings of the 27th Leeds\u2010Lyon Symposium on Tribology, G. Dalmaz et al., eds., Elsevier, Amsterdam, 2001, pp. 709\u2013719."},{"key":"R17","unstructured":"C. E. Goodyer, R. Fairlie, D. E. Hart, M. Berzins, and L. E. Scales,\n                      Calculation of friction in steady\u2010state and transient EHL simulations\n                      , in Transient Processes in Tribology, Proceedings of the 30th Leeds\u2010Lyon Symposium on Tribology, G. Dalmaz et al., eds., Elsevier, Amsterdam, 2004, pp. 579\u2013590."},{"key":"R18","unstructured":"C. E. Goodyer, J. Wood, and M. Berzins,\n                      A parallel grid based PSE for EHL problems\n                      , in Applied Parallel Computing, Proceedings of PARA\u201902, Lecture Notes in Comput. Sci. 2367, J. Fagerholm et al., eds., Springer\u2010Verlag, New York, 2002, pp. 523\u2013532."},{"key":"R19","volume-title":"Solving ordinary differential equations. I","author":"Hairer E.","year":"1993"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1002\/fld.881"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1115\/1.2920948"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/0912031"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2003.11.004"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1115\/1.3261260"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1115\/1.3261467"},{"key":"R26","unstructured":"Kyoung\u2010Sook Moon, Anders Szepessy, Ra\u00fal Tempone, Georgios Zouraris, Hyperbolic differential equations and adaptive numerics, Universitext, Springer, Berlin, 2001, 231\u20132801862903"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1115\/1.2833805"},{"key":"R28","first-page":"209","volume":"2","author":"Petrusevich A. I.","year":"1951","journal-title":"Izv. Akad. Nauk. SSSR (OTN)"},{"key":"R29","unstructured":"C. J. A. Roelands,\n                      Correlational Aspects of the Viscosity\u2010Temperature\u2010Pressure Relationship of Lubricating Oils\n                      , Ph.D. thesis, Technische Hogeschool Delft, Delft, The Netherlands, 1966."},{"key":"R30","unstructured":"L. E. Scales, J. E. Rycroft, N. R. Horswill, and B. P. Williamson,\n                      Simulation and observation of transient effects in elastohydrodynamic lubrication\n                      , SP\u20101182, in SAE International Fuels and Lubricants Meeting, Dearborn, MI, 1996, pp. 23\u201334."},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1137\/0901005"},{"key":"R32","volume-title":"Computer solution of ordinary differential equations","author":"Shampine L.","year":"1975"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2706-9"},{"key":"R34","unstructured":"F. Stenger,\n                      private communication\n                      , 2004."},{"key":"R35","volume-title":"Multigrid","author":"Trottenberg U.","year":"2001"},{"key":"R36","unstructured":"C. H. Venner,\n                      Multilevel Solution of the EHL Line and Point Contact Problems\n                      , Ph.D. thesis, University of Twente, Endschende, The Netherlands, 1991."},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1115\/1.2927329"},{"key":"R38","unstructured":"C. H. Venner and A. A. Lubrecht,\n                      Multilevel Methods in Lubrication\n                      , Elsevier, Amsterdam, 2000."},{"key":"R39","unstructured":"B. Watremetz, F. Colin, C. H. Venner, and A. A. Lubrecht,\n                      Optimum time step in a transient EHL contact\n                      , in Transient Processes in Tribology, Proceedings of the 30th Leeds\u2010Lyon Symposium on Tribology, G. Dalmaz et al., eds., Elsevier, Amsterdam, 2004, pp. 591\u2013600."},{"key":"R40","unstructured":"Y. Wijnant,\n                      Contact Dynamics in the Field of Elastohydrodynamic Lubrication\n                      , Ph.D. thesis, University of Twente, Endschende, The Netherlands, 1998."},{"key":"R41","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7225(86)90032-7"},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7225(87)90057-7"},{"key":"R43","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(87)90119-2"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/050622092","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:50:12Z","timestamp":1787334612000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/050622092"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":44,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/050622092"],"URL":"https:\/\/doi.org\/10.1137\/050622092","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}